Evaluating Alternative Operations Strategies to Improve Travel Time ...

SHRP 2 L11: Final Appendices
A. The new set of speed data parameters (after implementation of the treatment) is calculated
as follows:
Assumptions for the new scenario:
• The new average speed ( ) = 36.39 mph (the 2002 average speed)
V T
• The guaranteed speed (equivalent to the 2002 average speed) (I) = 36.39 mph
1
2
Standard deviation of ln speed or speed volatility ( α )= Σ(ln( S i
) −µ
)
N
α = 0.2094 for this example
The log-normal distribution using the log mean and the log standard deviation of the speed
is shown on Figure D.4 to confirm the speed data are log-normally distributed.
0.09
0.08
0.07
Figure D.4 - Log-Normal Distribution of the 2002 Speeds
Probability Density Function
0.06
0.05
0.04
0.03
0.02
0.01
0
0 10 20 30 40 50 60 70
Speed (mph)
Note: Weekends were excluded from the data.
B. Calculate the certainty-equivalent value for the after (2002) scenario.
The European Put option and a 5.00% risk-free interest rate are chosen.
Thus, r = 0.05
The 95 th percentile speed is now 27.91 mph. Therefore, the new contract length is given by:
Sigma is calculated using the formula for volatility in finance, which is the standard deviation
of the log speed divided by the square root of the contract length:
The intermediate and final option value calculations are performed:
2
ln( VT
/ I) + ( r+ σ / 2)( T −t)
d1
=
σ ( T − t)
SAMPLE PROBLEM-QUANTIFYING THE ECONOMIC BENEFIT OF IMPROVING TRAVEL-TIME RELIABILITY Page D-9